Optimal. Leaf size=83 \[ \frac{x \sqrt{c x}}{\sqrt [4]{a+b x^2}}+\frac{\sqrt{a} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0324452, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {314, 284, 335, 196} \[ \frac{x \sqrt{c x}}{\sqrt [4]{a+b x^2}}+\frac{\sqrt{a} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 314
Rule 284
Rule 335
Rule 196
Rubi steps
\begin{align*} \int \frac{\sqrt{c x}}{\sqrt [4]{a+b x^2}} \, dx &=\frac{x \sqrt{c x}}{\sqrt [4]{a+b x^2}}-\frac{1}{2} a \int \frac{\sqrt{c x}}{\left (a+b x^2\right )^{5/4}} \, dx\\ &=\frac{x \sqrt{c x}}{\sqrt [4]{a+b x^2}}-\frac{\left (a \sqrt [4]{1+\frac{a}{b x^2}} \sqrt{c x}\right ) \int \frac{1}{\left (1+\frac{a}{b x^2}\right )^{5/4} x^2} \, dx}{2 b \sqrt [4]{a+b x^2}}\\ &=\frac{x \sqrt{c x}}{\sqrt [4]{a+b x^2}}+\frac{\left (a \sqrt [4]{1+\frac{a}{b x^2}} \sqrt{c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )}{2 b \sqrt [4]{a+b x^2}}\\ &=\frac{x \sqrt{c x}}{\sqrt [4]{a+b x^2}}+\frac{\sqrt{a} \sqrt [4]{1+\frac{a}{b x^2}} \sqrt{c x} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0098727, size = 56, normalized size = 0.67 \[ \frac{2 x \sqrt{c x} \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )}{3 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{cx}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x}}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x}}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.05564, size = 44, normalized size = 0.53 \begin{align*} \frac{\sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} \Gamma \left (\frac{7}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x}}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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